Midbandgap electro-optic detection of Bloch oscillations

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Midbandgap electro-optic detection of Bloch oscillations

M. Fo¨rst,^*G. Segschneider, T. Dekorsy, and H. Kurz

Institut fu¨r Halbleitertechnik II, Rheinisch–Westfa¨lische Technische Hochschule Aachen, D-52056 Aachen, Germany

K. Ko¨hler

Fraunhofer-Institut fu¨r Angewandte Festko¨rperphysik, D-79108 Freiburg, Germany

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Received 8 September 1999; revised manuscript received 12 January 2000

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Bloch oscillations excited in a biased GaAs/Al_xGa_1⫺xAs superlattice are investigated in a time-resolved two-color electro-optic detection scheme. The detection of resonantly excited Bloch oscillations is based on the Pockels effect probed at a wavelength in the center of the band gap. The observed birefringence is induced by the macroscopic polarization of the electronic wave packets relative to the localized holes. The off-resonant detection away from optical transitions directly monitors the spatial dynamics of the electrons in amplitude and phase. The dependence of the amplitudes of the Bloch oscillating electrons on the applied electric fields are in good agreement with the electron-hole dipole lengths calculated by a quantum-mechanical model.

Bloch¹and Zener²have proposed in their theoretical studies on the dynamics of electronic wave packets in periodic potentials a temporal and spatial oscillation of the carriers in the presence of a static electric field. The frequency ␯ ^and the spatial amplitude L of these Bloch oscillations共BO’s兲are

␯⫽eFd/h and L⫽⌬/eF, 共1兲 respectively, where F is the applied electric field, d the po- tential period, h Planck’s constant, and ⌬ the electronic bandwidth. As proposed by Esaki and Tsu, this intriguing phenomenon can be realized in a biased semiconductor su- perlattice共SL兲.³ Here, BO’s are excited by the coherent su- perposition of several electronic Wannier-Stark 共WS兲 states^4,5with a common hole state by short laser pulses. The localization lengths of the electronic wave packets deviate from the semiclassical picture of BO’s 关Eq. 共1兲兴 if a fully quantum-mechanical calculation of the wave functions in the artificial semiconductor structure is used.⁶ Additionally, the amplitude of optically excited BO’s depends on the exciting laser pulse width and energy that determine the initial conditions of the wave packets.⁷

Pathbreaking investigations on BO’s have been performed in four-wave-mixing,^8,9THz emission,¹⁰ and resonant trans- mittive electro-optic sampling 共TEOS兲 experiments.¹¹In re- cent years, the experimental determination of the spatial amplitude of the oscillating electronic wave packets has become an objective of the studies. The amplitude determination of BO’s in a SL has been reported in a THz emission experiment.¹² The amplitudes were calculated from the detected THz radiation power by applying a superradiance theory. Subsequently, the BO amplitude has been carefully determined in spectrally and time-resolved four-wave mixing experiments.¹³ Using this technique, the dependence of the amplitudes on the spectral position of the exciting laser pulses was investigated.¹⁴ Resonant TEOS experiments performed with optical detection spectrally integrated over WS transitions can only provide a qualitative analysis of the electronic dynamics due to nonlinearities of the electro-optic effect at the interband resonances.

In this paper, we present the off-resonant electro-optic detection of Bloch oscillations in a GaAs/Al_xGa₁_⫺_xAs SL.

BO’s are excited resonantly, but detected electro-optically in the center of the band gap far away from optical transitions.

This detection scheme enables the direct osbervation of the spatial carrier dynamics in amplitude and phase. The BO amplitudes are obtained in dependence on the applied electric fields and compared to the semiclassical localization lengths of BO’s and a quantum-mechanical calculation of the oscillation amplitude of the electronic wave packets in the SL.

The investigated SL consists of 35 periods of 9.7 nm wide GaAs wells and 1.7 nm wide Al_0.3Ga_0.7As barriers. These are embedded between intrinsic Al_0.3Ga_0.7As buffer layers of 250 nm and 350 nm width resulting in an intrinsic region of 1 ␮m which is electro-optically active. The structure is grown on an共001兲oriented n-doped GaAs substrate and pre- pared as described in Ref. 11. The electronic miniband width is calculated to 18 meV via a Kronig-Penney model. The narrow heavy-hole miniband 共1.9 meV兲leads to a localization of heavy holes even at low electric fields.

The central wavelength of the laser pulses is chosen to excite electronic WS states and heavy holes. Since the heavy holes are assumed to be localized, the coherent and incoherent spatial dynamics of the electronic wave packets relative to the holes result in a dipole field P_z(␶,z) that leads to a voltage drop ⌬V_z(␶⁾⫽兰P_z(␶,z)dz in the SL. In first ap- proximation, the dipole field depends linearly on the relative displacement s_z(␶) between the distinct carriers and the longitudinal inhomogenous excitation density n_exc(z)

⬀exp(⫺␣^{z) by}

P共␶^,z兲⫽en_exc共z兲

⑀⑀0

s_z共␶兲, 共2兲

where ␣ is the absorption coefficient of the sample. These field dynamics can be detected via the electro-optic effect. In crystals of the 4¯ 2m point group, like III-V semiconductor SL’s, circularly polarized light propagating in the 共001兲 growth direction becomes elliptically polarized due to the

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birefringence that is induced by a longitudinal field. The de- gree of the ellipticity can be detected by the subtraction of the transmitted beam intensities along the directions of the index ellipsoid’s main axes共see, e.g., Refs. 11 and 15兲. The resulting TEOS signal depends linearly on the voltage drop

⌬V_z over the electro-optic active length:

⌬T_eo T₀ ⫽2␲

␭ n₀³r₆₃⌬V_z⫽:A_eo⫻⌬V_z. 共3兲 Here, T₀ and␭ are the transmitted intensity and the wave- length of the probe beam, n₀ the refractice index, and r₆₃the electro-optic coefficient of the crystal. A_eo summarizes all voltage independent factors. Isotropic changes of the refractive index, e.g., due to free carrier absorption,¹⁶ do not con- tribute to the anisotropic TEOS signal.

The measurements are performed in a two-color pump- probe setup at 10 K lattice temperature. The pump pulses of 125 fs are derived from a Ti:sapphire laser at a central wavelength of 800 nm. The laser beam is divided into one part for the excitation of the sample behind an external pulse com- pression and a second part to synchronously pump a lithium triborate 共LBO兲–based optical parametric oscillator 共OPO兲. The probe pulses derived from the OPO are tuned to a central wavelength of 1.49 ␮m with a length of 150 fs. Aniso- tropic transmission changes of the circularly polarized probe beam are detected behind a polarizing beamsplitter by two In_xGa₁_⫺_xAs photodiodes. The subtracted signal is sampled by a fast-scan data acquisition system 共Aixscan, GWU La- sertechnik, Germany兲.

Due to the lack of literature values for electro-optic coef- ficients at 10 K, it is incumbent to calibrate the anisotropic transmission changes to the voltage drop in the specific sample. Therefore, A_eo 关Eq. 共3兲兴 averaged over the spectral width of the probing light is measured in a quasi-cw 共conti- nous wave兲 TEOS experiment. Here, the anisotropic transmission changes are sampled phase-locked to the bias applied to the sample. The voltage is triangularly modulated from 0 V to⫺1 V reverse bias at a frequency of 4 kHz. The detected transmission changes are shown in the inset of Fig.

1. The signal linearly follows the applied voltage with a slope equal the coefficient A_eo⫽1.26⫻10^⫺⁴ V^⫺¹, which

will be used for the quantitative analysis of the time-resolved experiments. From this value and an effective refractive in- dex n₀⫽3.20 of the sample at 10 K, the effective electro- optic coefficient is calculated to be r₆₃*_⫽0.91⫻10^⫺¹² m/V.

This is in a good agreement with the room temperature value of r₆₃⫽1.50⫻10^⫺¹² m/V at 1.52 ␮m of bulk GaAs 共Ref.

17兲 and the temperature dependence that has been investigated down to lattice temperatures of 100 K.¹⁸ In our cali- bration with a modulation frequency of 4 kHz, we measure the unclamped electro-optic coefficient. In the time-resolved experiments with BO-induced electric-field modulations in the THz range, the clamped value has to be considered. With respect to room-temperature calculations of the clamped and unclamped values, we neglect the piezoelectrically induced lattice contributions to the electro-optic coefficient that are of opposite sign and assumed to be at least one order of magnitude smaller than those of the electronic subsystem.¹⁹

Figure 1 shows a time-resolved TEOS signal detected at a reverse bias of ⫺1.3 V and an excitation density of 2.7

⫻10⁹ cm^⫺²共in the first well兲. The central wavelength of the pump beam is located between the WS共⫺1兲and the WS共0兲 transitions. Initially, the detected signal consists of a step that follows the integral over the laser intensity profile. This step is attributed to the electric-field induced instantaneous displacement of the electronic wave packet relative to the localized holes within the pulse duration that creates an initial dipole field.²⁰ Additionally, the anisotropic transmission changes are periodically modulated due to the excitation of BO’s. These data directly visualize the coherent excitation of BO’s and the associated dipole field with a well defined phase relative to zero time delay. At excitation below the WS共0兲 transition, the electronic wave packets are initially displaced in the direction of the applied electric field. The cosinelike oscillation around this new ‘‘equilibrium’’ position starts in the same direction. This unidirectional move- ment of both components could not be extracted from THz emission, resonant TEOS, or four-wave mixing experiments that are not sensitive to the direction of the carrier motion at zero time delay. At larger time delays, the amplitude of the oscillation is damped and the negative background signal further increases. This effect results from the dephasing of the BOs that共i兲reduces the strength of the associated dipole field due to the reduced number of coherently oscillating electrons and 共ii兲 leads to a drift current of the scattered electrons through the SL which further screens the applied electric field.²¹ We should further note that the signals detected in resonant TEOS experiments are typically two or- ders of magnitude larger.¹¹ This enlargement is attributed to the resonant enhancement of the electro-optic effect at the WS transitions.

For further analysis, anisotropic transmission changes are detected at reverse biases increasing from ⫺0.9 V to

⫺1.6 V in steps of⫺0.1 V, i.e., in the WS regime of the SL. By folding the exciting laser spectrum with the spectral absorption of the sample, the excitation density was calculated to vary less than 8% in the relevant voltage range caused by the electric-field dependence of the absorption strengths of WS transitions. This fluctuation is comparably weak to our experimental error in the determination of the excitation density. The oscillatory signal contributions extracted numerically are shown in Fig. 2. An increase of the FIG. 1. Time-resolved TEOS signal at an excitation density of

2.7⫻10⁹ cm^⫺2

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in the first well

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for an applied reverse bias of

⫺1.3 V. The inset shows anisotropic transmission changes detected in a quasi-cw TEOS setup to determine the effective electro-optic coefficient of the investigated sample. The applied bias is modulated with a frequency of 4 kHz.

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oscillation frequency with increasing reverse bias is clearly visible manifesting the detection of BO’s. Additionally, the amplitude of the oscillations decreases with increasing reverse biases. Since the excitation density is approximately constant for the applied electric fields, the decrease of the amplitude is caused by a reduction of the BO induced dipole field due to the enhanced localization at higher fields. The dephasing time of the detected BOs determined from mono- exponentially decaying fits to the extracted oscillations increases from about 0.3 ps at⫺1.6 V to 0.5 ps at⫺0.9 V.

These values are below the longest dephasing times of 2.8 ps,¹¹but in the range of 0.37 ps,²²both observed in resonant TEOS experiments at comparable excitation densities, but in different samples. This discrepancy is attributed to different sample qualitities, e.g., regarding the interface roughness, that strongly influence the coherence dephasing time.

The frequencies of the BO’s obtained from Fourier transforms to the extracted oscillations are shown in the inset of Fig. 3. They increase linearly from 0.9 THz to 2.7 THz with increasing reverse biases. The slope obtained from a linear fit to the data is 2.5 THz/V which is in good agreement with the value expected from the given sample parameters by the semiclassical picture of BO’s. This is calculated to ed/hl_eo

⫽2.76 THz/V, assuming that the applied voltage drops over the whole intrinsic region of the sample. The accumulation of carriers at the SL boundary layers between successive pump pulses leads to a quasistatic screening of the applied electric field.²³Its amount of 6 kV/cm is independent on the applied bias and thus in agreement with the assumption of a constant excitation density for different applied fields.

In the following, the oscillation amplitudes of the electronic wave packets are determined as a function of the applied electric field. Therefore, the extracted oscillations are extrapolated to zero time delay by numerical fits. From the signal amplitude at␶⫽0, the spatial peak-to-peak amplitude of the oscillating electrons can be calculated considering the inhomogenous excitation density. The results are shown in Fig. 3 共squares兲 dependent on the electric fields that have been calculated from the peak positions of the Fourier transforms of the time-domain oscillations. The experimental errors mainly result from the determination of the excitation density. Error contributions due to the experimental equip- ment and the careful analysis of the time-domain data are of minor weight but not neglected. The obtained amplitudes

decrease from 21 nm at an electric field of 3.4 kV/cm to 6.5 nm at 9.6 kV/cm. Thus, the BO amplitudes are in the range between less than one and two SL periods in this electric- field range. Obviously, the semiclassical BO amplitudes 共dotted line兲 are about a factor of 2 larger than the experimentally detected amplitudes. This is in agreement with the results of Lyssenko et al. who quantified the oscillation am- plitudes in a 38 meV GaAs/Al_xGa₁_⫺_xAs superlattice.¹³ There, the wave packets reached about 2/3 of the semiclassical amplitude. The discrepancy between experimentally determined and semiclassical amplitudes are assumed to result from the precise excitation conditions, i.e., the spectral laser position, and the deformation of the wave functions due to different effective masses in barriers and wells. Recently, it has been shown that the excitation of electronic wave packets near the WS共0兲 transition leads to a reduced center of mass motion due to the wave packet’s distribution over a large part of the Brillouin zone with positive and negative masses.¹⁴ The associated breathing motion results in a reduced dipole field. Here, we compare the experimental results with a quantum-mechanical calculation of the electronic wave functions taking into account solely the difference of effective masses in the wells and barriers.⁶ The oscillation amplitudes are calculated in the assumption of the superpo- sition of the WS共⫺1兲 and the WS共0兲 states by a modified Numerov algorithm 共dashed line兲.²³ A comparison between the calculated and the experimental data yields a good agreement within the experimental errors.

In conclusion, we realized a two-color pump-probe setup for quantitative investigations of electric-field dynamics in III-V semiconductors with femtosecond time resolution. The field dynamics are detected electro-optically in the center of the bandgap enabling the direct visualization of spatial car- FIG. 2. Oscillatory contributions to the anisotropic transmission

changes at reverse biases increasing from⫺0.9 V to⫺1.6 V in steps of⫺0.1 V. For clarity, the traces are shifted vertically.

FIG. 3. Experimentally determined peak-to-peak amplitude of the oscillating electronic wave packets dependent on the electric fields

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squares

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. The dashed and dotted lines are calculations from a quantum-mechanical and the semiclassical theory, respectively. The inset shows the peak positions of Fourier transforms to the Bloch oscillations and a linear fit to the data.

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rier dynamics after optical excitation. This new method complements the established time-resolved techniques of four-wave mixing, THz emission and resonant TEOS with the advantage that共i兲the signal amplitudes can be calibrated, 共ii兲the detection is free from excitonic contributions, and the disadvantage of small signal amplitudes at the low excitation densities required for the observation of electronic coherence. Particularly, we monitored the space-time evolution of resonantly excited Bloch oscillations in a biased GaAs/Al_xGa₁_⫺_xAs superlattice. The initial displacement, the coherent oscillations, and the incoherent transport of elec-

tronic wave packets were clearly observed. The oscillation amplitudes of the electronic wave packets increase from less than one to about two superlattice periods in the investigated electric field range. The results can be modeled by quantum- mechanical calculations of the oscillation amplitudes in the superlattice.

The authors gratefully acknowledge G.C. Cho for stimu- lating suggestions and R. Martini for helpful discussions.

This work was supported by the Volkswagen Stiftung.

*Electronic address: foerst@iht-ii.rwth-aachen.de

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